In the case of Obergefell v. Hodges the US Supreme Court has been hearing final arguments regarding whether it should recognise Same-Sex Marriage as a "Right" under the US Constitution
The Court is considering 2 questions
(1) “Does the 14th amendment require a state to license a marriage between two people of the same sex?”
(2) “Does the 14th amendment require a state to recognize a marriage between two people of the same sex when their marriage was lawfully licensed and performed out-of-state?”
The relevant part of the 14th Amendment says
"No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws."
In addition to the 14th Amendment implicit in question(2) is Article IV, Section 1: of the US Constitution which says
"Full faith and credit shall be given in each state to the public acts, records, and judicial proceedings of every other state."
The US Supreme Court website provides transcripts of the legal argument and also audio recordings so you can read the legal argument on Question 1 here and Question 2 here: Or you can listen to an audio recording of the legal argument on Question 1here: and Question 2 here.
The SCOTUSblog provides a link to the (numerous) briefs and Amicus Curiae submissions made in the case HERE: as does the website of the American Bar Association
A fundamental argument frequently made by those who want the US Supreme Court to make Same-Sex marriage a Constitutional Right is the suggestion that banning Same-Sex marriage is similar to the earlier bans on Interacial Marriage which existed in many of the Southern States until these were overturned in the case of Loving v Virginia in 388 U.S. 1 (1967)
Interestingly at the outset of the legal argument between the parties US Chief Justice Roberts put his finger at the difference between this case and the Loving case when he said
"Well, you say join in the institution [ie Marriage]. The argument on the other side is that they're seeking to redefine the institution. Every definition that I looked up, prior to about a dozen years ago, defined marriage as unity between a man and a woman as husband and wife. Obviously, if you succeed, that core definition will no longer be operable.......... you're not seeking to join the institution, you're seeking to change what the institution is. The fundamental core of the institution is the oppositesex relationship and you want to introduce into it a samesex relationship"
Justice Kennedy similarly pointed the inherent problem of a decision of this significance being imposed on the United States by Judicial Fiat.
"This definition has been with us for millennia. And it it's very difficult for the Court to say, oh, well, we we know better."
What the Supreme Court will decide is difficult to predict but my personal view is that it will answer Question (1): No, and Question (2): Yes, on the basis both of the 14th Amendment and the principle of reciprocity in Article 1V. This answer will also avoid the danger of the Supreme Court being seen to impose its own subjective views and thereby overuling or sidelining the democratic process.